Abstract
We deduce an overcomplete free energy functional for D=1 particle systems with next neighbor interactions, where the set of redundant variables are the local block densities ϱ i of i interacting particles. The idea is to analyze the decomposition of a given pure system into blocks of i interacting particles by means of a mapping onto a hard rod mixture. This mapping uses the local activity of component i of the mixture to control the local association of i particles of the pure system. Thus it identifies the local particle density of component i of the mixture with the local block density ϱ i of the given system. Consequently, our overcomplete free energy functional takes on the hard rod mixture form with the set of block densities ϱ i representing the sequence of partition functions of the local aggregates of particle numbers i. The system of equations for the local particle density ϱ of the original system is closed via a subsidiary condition for the block densities in terms of ϱ. Analoguous to the uniform isothermal-isobaric technique, all our results are expressible in terms of effective pressures. We illustrate the theory with two standard examples, the adhesive interaction and the square-well potential. For the uniform case, our proof of such an overcomplete format is based on the exponential boundedness of the number of partitions of a positive integer (Hardy-Ramanujan formula) and on Varadhan's theorem on the asymptotics of a class of integrals.
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Tutschka, C., Cuesta, J.A. Overcomplete Free Energy Functional for D = 1 Particle Systems with Next Neighbor Interactions. Journal of Statistical Physics 111, 1125–1148 (2003). https://doi.org/10.1023/A:1023096031180
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DOI: https://doi.org/10.1023/A:1023096031180